Positive self-commutators of positive operators
Roman Drnovšek, Marko Kandić
TL;DR
The paper advances the understanding of positive self-commutators on Hilbert lattices by proving that every positive compact central operator on a separable infinite-dimensional Hilbert lattice is a self-commutator of a positive operator, and that every positive central operator is a sum of two positive self-commutators of positive operators. It develops constructive tools, notably diagonal-to-$X^*X$ representations with $X: ext{domain} o L^2[0,1]$ and corresponding $Y$ with $Y^2=XX^*$, and employs block-operator decompositions across isomorphic Hilbert-lattice models (e.g., $ ext{ell}^2$, $L^2[0,1]$, and their sums) to realize central operators as commutators. The work extends Radjavi-type order-analytic results to a lattice setting, clarifies when invertible central operators can or cannot be commutators, and provides a practical framework for expressing central operators as sums of self-commutators, with implications for the structure of operator algebras on Banach lattices.
Abstract
We consider a positive operator $A$ on a Hilbert lattice such that its self-commutator $C = A^* A - A A^*$ is positive. If $A$ is also idempotent, then it is an orthogonal projection, and so $C = 0$. Similarly, if $A$ is power compact, then $C = 0$ as well. We prove that every positive compact central operator on a separable infinite-dimensional Hilbert lattice $\mathcal H$ is a self-commutator of a positive operator. We also show that every positive central operator on $\mathcal H$ is a sum of two positive self-commutators of positive operators.